Tuesday, November 27, 2012

random points

This is a basic sketch that might provide a neat starting point for some randomized geometrical constructions. The inclusion of randomness can be helpful in the exploration of generality. When we want to analyze mathematical objects, we often choose handy examples that may or may not possess some quality that is in fact typical or possible. Just as a magician invites us to "pick a card, any card!" we might do well to "pick a triangle, any triangle!" or "pick a function, any function!" in order to gather observations that are less hampered by our own difficulties with constructing generality.

The randomness in this sketch relies on the animation of an independent point, which I am taking as sufficiently random. A point starts at the origin and it is animated for 1 second, squirming around 'randomly' according to Sketchpad's algorithm. Then I take the mantissa or fractional part of the x-coordinate of that point and multiply it by the size of the range we want. The range is user-defined by changing the xmin, xmax, ymin, ymax values that are visible when you click the Show Parameters button.

 Some things to try:
  • Trace the points to keep track of many trials. 
  • Define a function using the points. 
  • Construct a triangle using 3 points and look at the relationship between area and perimeter. 
  • Use some of the points and hide the others. 
  • Trace the midpoint of the segment between two random points. 

Download GSP5 file: randomPoints.gsp

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